Who should vote in a Primary?

Should people who are not members (registered voter as a member of the party) of a political party be allowed to vote for the person to lead the party. I have heard the argument that since the winner will be running for president then everyone should get to vote. If a person is registered as an Independent they are saying that they don't want to be associated with either major party so why should they get to decide who represent the party? To me it is like asking the people in your High School athletic conference to decide if you need a new school or not. Why should we allow people who don't want to be part of our group decide what we do?
We do not allow people who live outside of a school district, the city, the county, the state or even the country but pay taxes in that government entity to vote on issues that they will have to pay for.
My question is should primaries be open or closed and explain why?

I have always been for the theory that the more voices you have the better the country will be. But, when Rush had his ignorant followers try to change the democratic primaries in 2008 I have changed my mind. I hate to say that I think that they should be limited but in order to stop these ignorant right wing hate radio types from ruining elections you are better off to close elections to only those that are from that party. The ironic part is that the people that won't be able to vote in the primary are the ones that will actually decide the election.

You could let them vote on an independent ballot so they can express their choice but only the party ballots count.
I agree with you about people who try to influence the other parties primarys it is not right. That is why I would like the primaries closed and you have to be registered with a party more than a few days ahead of the primary.

Bob, I think that your idea would be the best for our system. But, it would be interesting to see if a party would pay attention to those ballots, and eventually those ballots become more important than the party ballot. I hate to agree with you that primaries should be closed, but it is the safest way to ensure people don't game the system.

Party primaries shouldn't exist. Parties should not exist. I'd like to see primaries as a weed out for presidential candidates. Primaries should be a free-for-all election to the ballot for multiple candidates regardless of party. Our system should not be boiled down to "this guy" vs. "that guy".

I think primaries should be closed and they are in a lot of states. But this is one of those situations where there's no right answer. Any point to it has an equally valid counter argument.
On one hand you stop party tampering. On the other you often get a candidate that appeals less to anyone who is undecided.
Most people generally agree more with one party than the other. So by having the primaries you get to vote for the candidate you prefer among those with similar views, either you're supporting your candidate through the whole process, or you're candidate loses but you still vote in party. It's not your first choice but 9 times out of 10 it's more desirable than the other party.
Then we get to independents. If we let them vote in both primaries then everyone starts going independent and we lose the benefit of a closed primary. However if a good enough candidate could get enough support a legitimate 3rd party could be formed.
This brings us to the final conundrum. We are stuck in this binary party system because a 3rd variable creates fear and doesn't get voted for. Most 3rd party candidates appeal to either the right or the left, but not both. When it gets to the voting booth a more appealing 3rd party candidate will often lose votes to second choice candidate representing an established party and already getting a greater deal of votes. Hence we end up with a 2 party system.
In the end it's really all a big magic trick, because the president is only 1/3rd of the legislative branch. While casting the most important vote from 1 person, he is still less significant than the 7 or so representatives each state sends to congress.